Download fly back converter

Module
3
DC to DC Converters
Version 2 EE IIT, Kharagpur 1
Lesson
22
Fly-Back Type Switched
Mode Power Supply
Version 2 EE IIT, Kharagpur 2
After completion of this lesson the reader will be able to:
(i)
(ii)
(iii)
(iv)
Identify the topology of a fly-back type switched mode power supply circuit.
Explain the principle of operation of fly-back SMPS circuit.
Calculate the ratings of devices and components used in fly-back converter for the
specified input and output voltages and for the required output power.
Design a simple fly-back converter circuit.
22.1 Introduction
Fly-back converter is the most commonly used SMPS circuit for low output power applications
where the output voltage needs to be isolated from the input main supply. The output power of
fly-back type SMPS circuits may vary from few watts to less than 100 watts. The overall circuit
topology of this converter is considerably simpler than other SMPS circuits. Input to the circuit
is generally unregulated dc voltage obtained by rectifying the utility ac voltage followed by a
simple capacitor filter. The circuit can offer single or multiple isolated output voltages and can
operate over wide range of input voltage variation. In respect of energy-efficiency, fly-back
power supplies are inferior to many other SMPS circuits but its simple topology and low cost
makes it popular in low output power range.
The commonly used fly-back converter requires a single controllable switch like,
MOSFET and the usual switching frequency is in the range of 100 kHz. A twoswitch topology exists that offers better energy efficiency and less voltage stress
across the switches but costs more and the circuit complexity also increases
slightly. The present lesson is limited to the study of fly-back circuit of single
switch topology.
22.2 Basic Topology of Fly-Back Converter
Fig.22.1 shows the basic topology of a fly-back circuit. Input to the circuit may be unregulated
dc voltage derived from the utility ac supply after rectification and some filtering. The ripple in
dc voltage waveform is generally of low frequency and the overall ripple voltage waveform
repeats at twice the ac mains frequency. Since the SMPS circuit is operated at much higher
frequency (in the range of 100 kHz) the input voltage, in spite of being unregulated, may be
considered to have a constant magnitude during any high frequency cycle. A fast switching
device (‘S’), like a MOSFET, is used with fast dynamic control over switch duty ratio (ratio of
ON time to switching time-period) to maintain the desired output voltage. The transformer, in
Fig.22.1, is used for voltage isolation as well as for better matching between input and output
voltage and current requirements. Primary and secondary windings of the transformer are wound
to have good coupling so that they are linked by nearly same magnetic flux. As will be shown in
the next section the primary and secondary windings of the fly-back transformer don’t carry
current simultaneously and in this sense fly-back transformer works differently from a
normal transformer. In a normal transformer, under load, primary and secondary windings
conduct simultaneously such that the ampere turns of primary winding is nearly balanced by the
opposing ampere-turns of the secondary winding (the small difference in ampere-turns is
required to establish flux in the non-ideal core). Since primary and
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N1:N2
D
Edc
C
Primary Side
Load
VO
Switch S
Gate pulses
Fig. 22.1 Fly Back Converter
secondary windings of the fly-back transformer don’t conduct simultaneously they are more like
two magnetically coupled inductors and it may be more appropriate to call the fly-back
transformer as inductor-transformer. Accordingly the magnetic circuit design of a fly-back
transformer is done like that for an inductor. The details of the inductor-transformer design are
dealt with separately in some later lesson. The output section of the fly-back transformer, which
consists of voltage rectification and filtering, is considerably simpler than in most other switched
mode power supply circuits. As can be seen from the circuit (Fig.22.1), the secondary winding
voltage is rectified and filtered using just a diode and a capacitor. Voltage across this filter
capacitor is the SMPS output voltage.
It may be noted here that the circuit shown in Fig.22.1 is rather schematic in nature. A more
practical circuit will have provisions for output voltage and current feedback and a controller for
modulating the duty ratio of the switch. It is quite common to have multiple secondary windings
for generating multiple isolated voltages. One of the secondary outputs may be dedicated for
estimating the load voltage as well as for supplying the control power to the circuit. Further, as
will be discussed later, a snubber circuit will be required to dissipate the energy stored in the
leakage inductance of the primary winding when switch ‘S’ is turned off.
Under this lesson, for ease of understanding, some simplifying assumptions are made. The
magnetic circuit is assumed to be linear and coupling between primary and secondary windings
is assumed to be ideal. Thus the circuit operation is explained without consideration of winding
leakage inductances. ON state voltage drops of switches and diodes are neglected. The windings,
the transformer core, capacitors etc. are assumed loss-less. The input dc supply is also assumed
to be ripple-free.
[A brief idea of a more practical fly-back converter will be given towards the end of this
lesson.]
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22.3 Principle of Operation
During its operation fly-back converter assumes different circuit-configurations. Each of these
circuit configurations have been referred here as modes of circuit operation. The complete
operation of the power supply circuit is explained with the help of functionally equivalent
circuits in these different modes.
As may be seen from the circuit diagram of Fig.22.1, when switch ‘S’ is on, the primary winding
of the transformer gets connected to the input supply with its dotted end connected to the positive
side. At this time the diode ‘D’ connected in series with the secondary winding gets reverse
biased due to the induced voltage in the secondary (dotted end potential being higher). Thus with
the turning on of switch ‘S’, primary winding is able to carry current but current in the secondary
winding is blocked due to the reverse biased diode. The flux established in the transformer core
and linking the windings is entirely due to the primary winding current. This mode of circuit has
been described here as Mode-1 of circuit operation. Fig. 22.2(a) shows (in bold line) the current
carrying part of the circuit and Fig. 22.2(b) shows the circuit that is functionally equivalent to the
fly-back circuit during mode-1. In the equivalent circuit shown, the conducting switch or diode is
taken as a shorted switch and the device that is not conducting is taken as an open switch. This
representation of switch is in line with our assumption where the switches and diodes are
assumed to have ideal nature, having zero voltage drop during conduction and zero leakage
current during off state.
Vpri = Edc , Vsec = Edc*N2/N1
+
Edc
VO
+
Vpri
Edc
Vsec VO
N1 : N2
Fig.22.2(a): Current path during Mode-1 of
circuit operation
Fig.22.2(b): Equivalent circuit in Mode-1
Under Mode-1, the input supply voltage appears across the primary winding inductance and the
primary current rises linearly. The following mathematical relation gives an expression for
current rise through the primary winding:
d
EDC = LPr i × iPr i ------------------------------------------------------------(22.1),
dt
where EDC is the input dc voltage, LPr i is inductance of the primary winding and iPri is
the instantaneous current through primary winding.
Linear rise of primary winding current during mode-1 is shown in Fig.22.5(a) and Fig.22.5(b).
As described later, the fly-back circuit may have continuous flux operation or discontinuous flux
operation. The waveforms in Fig.22.5(a) and Fig.22.5(b) correspond to circuit operations in
continuous and discontinuous flux respectively. In case the circuit works in continuous flux
mode, the magnetic flux in the transformer core is not reset to zero before the next cyclic turning
ON of switch ‘S’. Since some flux is already present before ‘S’ is turned on, the primary winding
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current in Fig. 22.3(a) abruptly rises to a finite value as the switch is turned on. Magnitude of the
current-step corresponds to the primary winding current required to maintain the previous flux in
the core.
At the end of switch-conduction (i.e., end of Mode-1), the energy stored in the magnetic field of
the fly back inductor-transformer is equal to LPr i I P2 2 , where I P denotes the magnitude of
primary current at the end of conduction period. Even though the secondary winding does not
conduct during this mode, the load connected to the output capacitor gets uninterrupted current
due to the previously stored charge on the capacitor. During mode-1, assuming a large capacitor,
the secondary winding voltage remains almost constant and equals to VSec = EDC × N 2 / N1 .
During mode-1, dotted end of secondary winding remains at higher potential than the other end.
Under this condition, voltage stress across the diode connected to secondary winding (which is
now reverse biased) is the sum of the induced voltage in secondary and the output voltage
( Vdiode = VO + EDC × N 2 / N1 ).
Mode-2 of circuit operation starts when switch ‘S’ is turned off after conducting for some time.
The primary winding current path is broken and according to laws of magnetic induction, the
voltage polarities across the windings reverse. Reversal of voltage polarities makes the diode in
the secondary circuit forward biased. Fig. 22.3(a) shows the current path (in bold line) during
mode-2 of circuit operation while Fig. 22.3(b) shows the functional equivalent of the circuit
during this mode.
Vpri = VO*N1/N2 , Vsec= VO
+
Edc
VO
+
Edc
Vpri
Vsec VO
N1: N2
Fig:22.3(a) : Current path during Mode-2 of
circuit operation
Fig.22.3(b): Equivalent circuit in Mode-2
In mode-2, though primary winding current is interrupted due to turning off of the switch ‘S’, the
secondary winding immediately starts conducting such that the net mmf produced by the
windings do not change abruptly. (mmf is magneto motive force that is responsible for flux
production in the core. Mmf, in this case, is the algebraic sum of the ampere-turns of the
two windings. Current entering the dotted ends of the windings may be assumed to
produce positive mmf and accordingly current entering the opposite end will produce
negative mmf.) Continuity of mmf, in magnitude and direction, is automatically ensured as
sudden change in mmf is not supported by a practical circuit for reasons briefly given below.
[mmf is proportional to the flux produced and flux, in turn, decides the energy stored in
the magnetic field (energy per unit volume being equal to B 2 2μ , B being flux per unit
area and μ is the permeability of the medium). Sudden change in flux will mean sudden
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change in the magnetic field energy and this in turn will mean infinite magnitude of
instantaneous power, some thing that a practical system cannot support.]
For the idealized circuit considered here, the secondary winding current abruptly rises from zero
to I P N1 N 2 as soon as the switch ‘S’ turns off. N 1 and N 2 denote the number of turns in the
primary and secondary windings respectively. The sudden rise of secondary winding current is
shown in Fig. 22.5(a) and Fig. 22.5(b). The diode connected in the secondary circuit, as shown in
Fig.22.1, allows only the current that enters through the dotted end. It can be seen that the
magnitude and current direction in the secondary winding is such that the mmf produced by the
two windings does not have any abrupt change. The secondary winding current charges the
output capacitor. The + marked end of the capacitor will have positive voltage. The output
capacitor is usually sufficiently large such that its voltage doesn’t change appreciably in a single
switching cycle but over a period of several cycles the capacitor voltage builds up to its steady
state value.
The steady-state magnitude of output capacitor voltage depends on various factors, like,
input dc supply, fly-back transformer parameters, switching frequency, switch duty ratio
and the load at the output. Capacitor voltage magnitude will stabilize if during each switching
cycle, the energy output by the secondary winding equals the energy delivered to the load.
As can be seen from the steady state waveforms of Figs.22.5(a) and 22.5(b), the secondary
winding current decays linearly as it flows against the constant output voltage (VO). The linear
d
decay of the secondary current can be expressed as follows: LSec × iSec = −VO ---------- (22.2),
dt
Where, LSec and iSec are secondary winding inductance and current respectively.
VO is the stabilized magnitude of output voltage.
Under steady-state and under the assumption of zero on-state voltage drop across diode, the
secondary winding voltage during this mode equals VO and the primary winding voltage =
VON1/N2 (dotted ends of both windings being at lower potential). Under this condition, voltage
stress across switch ‘S’ is the sum total of the induced emf in the primary winding and the dc
supply voltage (Vswitch = EDC + VON1/N2).
The secondary winding, while charging the output capacitor (and feeding the load), starts
transferring energy from the magnetic field of the fly back transformer to the power supply
output in electrical form. If the off period of the switch is kept large, the secondary current gets
sufficient time to decay to zero and magnetic field energy is completely transferred to the output
capacitor and load. Flux linked by the windings remain zero until the next turn-on of the switch,
and the circuit is under discontinuous flux mode of operation. Alternately, if the off period of the
switch is small, the next turn on takes place before the secondary current decays to zero. The
circuit is then under continuous flux mode of operation.
During discontinuous mode, after complete transfer of the magnetic field energy to the output,
the secondary winding emf as well as current fall to zero and the diode in series with the winding
stops conducting. The output capacitor however continues to supply uninterrupted voltage to the
load. This part of the circuit operation has been referred to as Mode-3 of the circuit operation.
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Mode-3 ends with turn ON of switch ‘S’ and then the circuit again goes to Mode-1 and the
sequence repeats.
+
Edc
VO
Fig:22.4 (a) : Current flow path during Mode3 of circuit operation
+
Edc
VO
Fig.22.4(b): Equivalent circuit in Mode-3
Figs.22.4(a) and 22.4(b) respectively show the current path and the equivalent circuit during
mode-3 of circuit operation. Figs.22.5(a) and 22.5(b) show, the voltage and current waveforms of
the winding over a complete cycle. It may be noted here that even though the two windings of
the fly-back transformer don’t conduct simultaneously they are still coupled magnetically
(linking the same flux) and hence the induced voltages across the windings are proportional to
their number of turns.
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IP
Io
I pri
0
tON
T
IP X N1 / N2
Time
Io X N1 / N2
I sec
0
T
tON
Time
EDC
V pri
VO X N1 / N2
0
tON
MODE-1
V load
T
MODE-2
Time
MODE-1
VO
Time
Fig.22.5(a): Fly-back circuit waveforms under continuous magnetic flux
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22.4 Circuit Equations Under Continuous-Flux Operation
The waveforms in Fig.22.5(a) correspond to steady state operation under continuous magnetic
flux. ‘tON’ denotes the time for which the fly-back switch is ON during each switching cycle. ‘T’
stands for the time period of the switching cycle. The ratio ( tON /T) is known as the duty cycle
(δ) of the switch. As can be seen from Fig.22.5(a), the primary winding current rises from I0 to IP
in ‘δT’ time. In terms of input supply voltage (Edc) and the primary winding inductance ( LPr i )
the following relation holds:
(IP - I0) = (Edc / LPr i ) δT ------------------------------------------(22.3),
Under steady state the energy input to primary winding during each ON duration equals: 0.5Edc
(IP + I0) δT and similarly the output energy in each cycle equals V0 ILoad T, where V0 is the
output voltage magnitude and ILoad denotes the load current. Equating energy input and energy
output of the converter (the converter was assumed loss-less) in each supply cycle, one gets:
0.5Edc (IP + I0) δ = V0 ILoad ------------------------------------------(22.4),
The mean (dc) voltage across both primary and secondary windings must be zero under every
steady state. When the switch is ON, the primary winding voltage equals input supply voltage
and when the switch is OFF the reflected secondary voltage appears across the primary winding.
Under the assumption of ideal switch and diode,
Edc δ = (N1 / N2) V0 (1-δ) ------------------------------------------(22.5),
where N1 and N2 are the number of turns in primary and secondary windings and (N1/N2)V0 is
the reflected secondary voltage across the primary winding (dotted end of the windings at lower
potential) during mode-2 of circuit operation.
One needs to know the required ratings for the switch and the diode used in the converter. When
the switch is OFF, it has to block a voltage (Vswitch) that equals to the sum of input voltage and
the reflected secondary voltage during mode-2.
Thus, Vswitch = Edc + (N1 / N2) V0 -----------------------------------(22.6),
When the switch in ON, the diode has to block a voltage (Vdiode) that equals to the sum of output
voltage and reflected primary voltage during mode-1, i.e.,
Vdiode = V0 + Edc (N2 / N1) -----------------------------------(22.6a)
Since the intended switching frequency for SMPS circuits is generally in the range of 100kHz,
the switch and the diode used in the fly-back circuit must be capable of operating at high
frequency. The switch and the transformer primary winding must be rated to carry a repetitive
peak current equal to IP (related to maximum output power as given by Eqns. 22.3 to 22.5).
Similarly the secondary winding and the diode put in the secondary circuit must be rated to carry
a repetitive peak current equal to the maximum expected load current. The magnetic core of the
high frequency inductor-transformer must be chosen properly such that the core does not saturate
even when the primary winding carries the maximum expected current. Also, the transformer
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core (made of ferrite material) must have low hysteresis loss even at high frequency operation.
Since the ferrite cores have very low conductivity, the eddy current related loss in the core is
generally insignificant.
22.5 Circuit Equations Under Discontinuous-Flux Mode
Fig. 22.5(b) shows some of the important voltage and current waveforms of the fly-back circuit
when it is operating in the discontinuous flux mode. During mode-3 of the circuit operation,
primary and secondary winding currents as well as voltages are zero. The load, however,
continues to get a reasonably steady voltage due to the relatively large output filter capacitor.
IP
I pri
0
tON
T
Time
T
Time
IP X N1 / N2
I sec
0
tON
EDC
V pri
VO X N1 / N2
0
tON
MODE-1
Time
T
MODE-2
MODE-3
MODE-1
VO
V load 0
Time
Fig.22.5(b): Fly-back circuit waveforms under discontinuous flux
With the turning ON of the switch, the primary winding current starts building up linearly from
zero and at the end of mode-1 the magnetic field energy due to primary winding current rises to
1
L pri I P2 . This entire energy is transferred to the output at the end of mode-2 of circuit operation.
2
Under the assumption of loss-less operation the output power (Po) can be expressed as:
Po =
1
L pri I P2 fswitch -----------------------------------(22.7),
2
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where fswitch (=1/T) is the switching frequency of the converter.
It may be noted that output power Po is same as ‘V0 ILoad’ used in Eqn.(22.4). The volt-time area
equation as given in Eqn.(22.5) gets modified under discontinuous flux mode of operation as
follows:
Edc δ ≤ (N1 / N2) V0 (1-δ) ------------------------------------------(22.8)
Average voltage across windings over a switching cycle is still zero. The inequality sign of
Eqn.22.8 is due to the fact that during part of the OFF period of the switch [= (1-δ)T], the
winding voltages are zero. This zero voltage duration had been identified earlier as mode-3 of the
circuit operation. The equality sign in Eqn.(22.8) will correspond to just-continuous case, which
is the boundary between continuous and discontinuous mode of operation. The expression for
Vswitch and Vdiode, as given in Eqns.(22.6) and (22.6a), will hold good in discontinuous mode also.
22.6 Continuous Versus Discontinuous Flux Mode of Operation
A practical fly-back type SMPS circuit will have a closed loop control circuit for output voltage
regulation. The controller modulates the duty ratio of the switch to maintain the output voltage
within a small tolerable ripple voltage band around the desired output value. If the load is very
light, very small amount of energy needs to be input to the circuit in each switching cycle. This is
achieved by keeping the ON duration of the switch low, resulting in low duty ratio (δ). Within
this small ON time only a small amount of current builds up in the primary winding. The off
duration of the switch, which is (1-δ) fraction of the switching time period, is relatively large.
Mode-2 duration of the circuit operation is also small as the magnetic field energy is quickly
discharged into the output capacitor. Thus, at light load, the circuit is in mode-3 for significant
duration. As the load increases the mode-3 duration, during which there is zero winding currents
and zero flux through the core, reduces and the circuit is driven towards continuous flux mode.
The circuit operation changes from discontinuous to continuous flux mode if the output power
from the circuit increases beyond certain value. Similarly if the applied input voltage decreases,
keeping the load power and switching frequency constant, the circuit tends to go in continuous
flux mode of operation.
For better control over output voltage, discontinuous flux mode of operation is preferred.
However, for the given transformer and switch ratings etc., more output power can be transferred
during continuous flux mode. A common design thumb rule is to design the circuit for operation
at just-continuous flux mode at the minimum expected input voltage and at the maximum (rated)
output power.
22.7 A Practical Fly-Back Converter
The fly-back converter discussed in the previous sections neglects some of the practical aspects
of the circuit. The simplified and idealized circuit considered above essentially conveys the basic
idea behind the converter. However a practical converter will have device voltage drops and
losses, the transformer shown will also have some losses. The coupling between the primary and
secondary windings will not be ideal. The loss part of the circuit is to be kept in mind while
designing for rated power. The designed input power (Pin) should be equal to Po/η, where Po is
the required output power and η is the efficiency of the circuit. A typical figure for η may be
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taken close to 0.6 for first design iteration. Similarly one needs to counter the effects of the nonideal coupling between the windings. Due to the non-ideal coupling between the primary and
secondary windings when the primary side switch is turned-off some energy is trapped in the
leakage inductance of the winding. The flux associated with the primary winding leakage
inductance will not link the secondary winding and hence the energy associated with the leakage
flux needs to be dissipated in an external circuit (known as snubber). Unless this energy finds a
path, there will be a large voltage spike across the windings which may destroy the circuit.
Fig.22.6 shows a practical fly-back converter. The snubber circuit consists of a fast recovery
diode in series with a parallel combination of a snubber capacitor and a resistor. The leakageinductance current of the primary winding finds a low impedance path through the snubber diode
to the snubber capacitor. It can be seen that the diode end of the snubber capacitor will be at
higher potential. To check the excessive voltage build up across the snubber capacitor a resistor
is put across it. Under steady state this resistor is meant to dissipate the leakage flux energy. The
power lost in the snubber circuit reduces the overall efficiency of the fly-back type SMPS circuit.
A typical figure for efficiency of a fly-back circuit is around 65% to 75%. In order that snubber
capacitor does not take away any portion of energy stored in the mutual flux of the windings, the
minimum steady state snubber capacitor voltage should be greater than the reflected secondary
voltage on the primary side. This can be achieved by proper choice of the snubber-resistor and
by keeping the RC time constant of the snubber circuit significantly higher than the switching
time period. Since the snubber capacitor voltage is kept higher than the reflected secondary
voltage, the worst-case switch voltage stress will be the sum of input voltage and the peak
magnitude of the snubber capacitor voltage.
Edc
RS
S
N
U
B
B
E
R
N1:N2
D
C
Load
V (o/p)
N3
PWM
Control
Block
Current Feedback
Fig. 22.6 A Practical Fly Back Converter
The circuit in Fig.22.6 also shows, in block diagram, a Pulse Width Modulation (PWM) control
circuit to control the duty ratio of the switch. In practical fly-back circuits, for closed loop output
voltage regulation, one needs to feed output voltage magnitude to the PWM controller. In order
to maintain ohmic isolation between the output voltage and the input switching circuit the output
voltage signal needs to be isolated before feeding back. A popular way of feeding the isolated
voltage information is to use a tertiary winding. The tertiary winding voltage is rectified in a way
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similar to the rectification done for the secondary winding. The rectified tertiary voltage will be
nearly proportional to the secondary voltage multiplied by the turns-ratio between the windings.
The rectified tertiary winding voltage also doubles up as control power supply for the PWM
controller. For initial powering up of the circuit the control power is drawn directly from the
input supply through a resistor (shown as RS in Fig.22.6) connected between the input supply
and the capacitor of the tertiary circuit rectifier. The resistor ‘RS’ is of high magnitude and causes
only small continuous power loss.
In case, multiple isolated output voltages are required, the fly-back transformer will need to have
multiple secondary windings. Each of these secondary winding voltages are rectified and filtered
separately. Each rectifier and filter circuit uses the simple diode and capacitor as shown earlier
for a single secondary winding. In the practical circuit shown above, where a tertiary winding is
used for voltage feedback, it may not be possible to compensate exactly for the secondary
winding resistance drop as the tertiary winding is unaware of the actual load supplied by the
secondary winding. However for most applications the small voltage drop in the winding
resistance may be tolerable. Else, one needs to improve the voltage regulation by adding a linear
regulator stage in tandem (as mentioned in Chapter-21) or by giving a direct output voltage
feedback to the control circuit.
Quiz Problems
(i) What kind of output rectifier and filter circuit is used in a fly back converter?
(a) a four-diode bridge rectifier followed by a capacitor
(b) a single diode followed by an inductor-capacitor filter
(c) a single diode followed by a capacitor
(d) will require a center-tapped secondary winding followed by a full wave rectifier and
a output filter capacitor.
(ii) A fly-back converter operates in discontinuous conduction mode with fixed ON duration of
the switch in each switching cycle. Assuming input voltage and the resistive load at the
output to remain constant, how will the output voltage change with change in switching
frequency? (Assume discontinuous conduction through out and neglect circuit losses.)
(a) Output voltage varies directly with switching frequency.
(b) Output voltage varies inversely with switching frequency.
(c) Output voltage varies directly with square root of switching frequency.
(d) Output voltage is independent of switching frequency.
(iii) A fly-back converter has primary to secondary turns ratio of 15:1. The input voltage is
constant at 200 volts and the output voltage is maintained at 18 volts. What should be the
snubber capacitor voltage under steady state?
(a)
(b)
(c)
(d)
More than 270 volts.
More than 200 volts but less than 270 volts.
Less than 18 volts.
Not related to input or output voltage.
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(iv) A fly-back converter is to be designed to operate in just-continuous conduction mode when
the input dc is at its minimum expected voltage of 200 volts and when the load draws
maximum power. The load voltage is regulated at 16 volts. What should be the primary to
secondary turns ratio of the transformer if the switch duty ratio is limited to 80%. Neglect
ON-state voltage drop across switch and diodes.
(a)
(b)
(c)
(d)
20 :1
30 :1
25 :2
50 :1
Answers to quiz problems: (i-c), (ii-c), (iii-a), (iv-d).
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